Question

which z-values correspond to the middle 70% of the standard normal distribution? round your answers to the nearest thousandth. < z <

Explanation:

Step1: Find the area in the tails

Since the middle area is \(70\% = 0.7\), the area in the two tails is \(1 - 0.7=0.3\). The area in each tail is \(\frac{0.3}{2}=0.15\).

Step2: Find the \(z\) - value for the left - hand tail

We want to find \(z\) such that \(P(Z < z)=0.15\). Using a standard normal table (or a calculator with a normal - distribution function, e.g., the inverse of the cumulative distribution function for the standard normal distribution \(\Phi^{-1}(p)\)), we find that \(z\approx - 1.036\) (because \(\Phi(-1.036)\approx0.15\)).

Step3: Use symmetry

Because the standard normal distribution is symmetric about \(z = 0\), if \(z_1\) is the \(z\) - value for the left - hand tail, then the \(z\) - value for the right - hand tail \(z_2=-z_1\). So \(z_2 = 1.036\)

Answer:

\(-1.036